To calculate the new coordinates after rotation:
\[ X = x \cos(\theta) + y \sin(\theta) \]
\[ Y = -x \sin(\theta) + y \cos(\theta) \]
Coordinate rotation involves rotating a point in the x-y plane by a specified angle about the origin. This is commonly used in various fields such as computer graphics, robotics, and physics to transform coordinates and analyze motion or orientation.
Let's assume the following values:
Step 1: Convert the angle to radians:
\[ \theta = 45 \times \frac{\pi}{180} = 0.7854 \text{ radians} \]
Step 2: Calculate the new X coordinate:
\[ X = 3 \cos(0.7854) + 4 \sin(0.7854) = 4.95 \]
Step 3: Calculate the new Y coordinate:
\[ Y = -3 \sin(0.7854) + 4 \cos(0.7854) = 0.71 \]
Let's assume the following values:
Step 1: Convert the angle to radians:
\[ \theta = 90 \times \frac{\pi}{180} = 1.5708 \text{ radians} \]
Step 2: Calculate the new X coordinate:
\[ X = 5 \cos(1.5708) + 6 \sin(1.5708) = 6 \]
Step 3: Calculate the new Y coordinate:
\[ Y = -5 \sin(1.5708) + 6 \cos(1.5708) = -5 \]
